Two generalizations of the nonabelian tensor product

TL;DR

This paper introduces the box-tensor product as a generalization of the nonabelian tensor product, extending finiteness results and proving new finiteness theorems for related tensor products and homology groups.

Contribution

It defines the box-tensor product, extends finiteness results to it, and proves new finiteness theorems for Inassaridze's tensor product and nonabelian homology groups.

Findings

Finiteness of the box-tensor product when factors are finite

Finiteness of Inassaridze's tensor product under new conditions

Finiteness of low-dimensional nonabelian homology groups

Abstract

The purpose of this paper is two fold. First we introduce the box-tensor product of two groups as a generalization of the nonabelian tensor product of groups. We extend various results for nonabelian tensor products to the box-tensor product such as the finiteness of the product when each factor is finite. This would give yet another proof of Ellis's theorem on the finiteness of the nonabelian tensor product of groups when each factor is finite. Secondly, using the methods developed in proving the finiteness of the box-tensor product, we prove the finiteness of Inassaridze's tensor product under some additional hypothesis which generalizes his results on the finiteness of his product. In addition, we prove an Ellis like finiteness theorem under weaker assumptions, which is a generalization of his theorem on the finiteness of nonabelian tensor product. As a consequence, we prove the finiteness of low-dimensional nonabelian homology groups.